# Linear Function

## Linear Function

Linear Function is defined as the real valued function $f : R \rightarrow R$ , y = f(x) = mx + c for each $x \in R$ where m and c is a constant

y = x+ 2 y=2x -3 y=x-2 y =4x The above all are example of linear function
Domain and Range of the Linear Function

For $f : R \rightarrow R , y = f(x) = mx + c for each$x \in R$Domain = R Range = R Graph of the Linear Function We can draw the graph on the Cartesian plan with value of x on the x-axis and value of y=f(x) on the y-axis. We can plot the point and join the point to obtain the graph. Here in case of the constant function,the graph will be a straight line $m$ is the slope and $b$ is the $y$ intercept. If $m$ is positive then the line rises to the right and if $m$ is negative then the line falls to the right Graph for the linear function with m and c both positive Graph for the linear function with positive m and negative c Graph for the linear function with m and c both negative Graph for the linear function with negative m and positive c Graph for the linear function with positive m and c=0. This passes through origin. Such type of linear function is also called proportional function Graph for the linear function with negative m and c=0. This passes through origin Identify Function and constant function are special cases of Linear function if m =1 and c=0, Linear function becomes f(x) =x which is a identity function If m=0 ,then Linear function becomes f(x) =c which is a Constant function ## Solved examples of Linear Functions 1. which is below function is a Linear function? a.$y =2x$b.$y = 11 -x$c.$ y= \frac {2}{3} x + \frac {1}{4} $d.$ x^2 + y^2=1$e.$y =x^3$f.$y =x^2 +1$Solution For the function to be a Linear function ,it should be of the form (mx+c) a. This is Linear function as of the form (mx+c) b. This is Linear function as of the form (mx+c) c. This is Linear function as of the form (mx+c) d. This is not a linear function e. This is not a linear function f. This is not a linear function 2. which of the graph represent Linear function? Solution The graph should be straight line for the function to be constant function So C and D are constant function 3. Let f = {(1,1), (2,3), (0, -1), (-1, -3)} be a linear function from Z into Z. Find f(x). Solution Since f is a linear function, f (x) = mx + c. Also, since$(1, 1), (0, - 1) \in Function$, f (1) = m + c = 1 and f (0) = c = -1. This gives m = 2 and so,f(x) = 2x - 1. ### Quiz Time Question 1The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by$t(C) =\frac {9}{5} C + 32$Which of the following is incorrect? A.t(0)=32 B. t(-5)=23 C. t(10)=48 D. None of the above Question 2If a function is defined as$f = {(x, y) | y = 2x + 7, \; where \; x \in R \; and \; -5 \leq x \leq 5}$is a relation. Then find the domain and Range of Function? A. Domain=[-5,5], range=[3,17] B. Domain=[-5,5], range=[-3,17] C. Domain=[-5,5], range=[-3,-17] D. Domain=[-5,5], range=[-5,5] Question 3 Let f = {(1,1), (2,3), (3,5), (4,7)} be a linear function from Z into Z. and f(x) =px +q then A. p=1,q=1 B. p=1,q=2 C. p=2,q=1 D. p=2,q=-1 Question 4 Let$f(x) =c $,find the value of$f(2) -f(1)$A. 4 B. 2 C. 0 D. 1 Question 5 The slope of the Linear function y=11x-1 is A. 0 B. 11 C. -1 D. None of these Question 6Find the Range and domain of the function$f(x) =x +2\$
A. Domain = R, Range =R
B. Domain = R - {2}, Range = R
C. Domain = R , Range = R -{2}
D. Domain = R - {1}, Range = R